Incorporating high-accuracy, finite-volume shock stabilization methods into a quantum algorithm for nonlinear partial differential equations

Conservation laws are of fundamental importance in science and engineering, and so are ubiquitous in applications. For continuum systems such as classical fluids and plasmas, they give rise to coupled systems of nonlinear partial differential equations (PDEs) whose solutions can develop discontinuities such as shocks. Finite-volume (FV) methods are an important tool for finding approximate solutions to such PDEs as they are based on an integral formulation of the conservation laws that can handle the presence of discontinuities. Weighted essentially nonoscillatory (WENO) methods are a class of high-resolution methods that can (i) produce solutions of arbitrarily high-order accuracy in regions where the solution is smooth, while (ii) simultaneously quashing the development of nonphysical oscillations in the vicinity of discontinuities that often cause numerical simulations to fail. In this paper we show how FV and WENO methods can be incorporated into a quantum algorithm for solving systems of nonlinear PDEs (QPDE) introduced by one of the authors. We numerically simulate application of the modified QPDE algorithm to three verification problems. The first has a smooth solution, while the other two are well-known one-dimensional Riemann problems whose exact solutions contain moving shocks, contact discontinuities, and rarefaction waves. For all three problems excellent agreement is found between the results of the QPDE simulations and the exact solutions.